1.15.2: Heat Capacities: Solutions: Solutes: Interaction Parameters
We describe an excess enthalpy \(\mathrm{H}^{\mathrm{E}}\) for a solution prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of solute \(j\) (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) ion terms of solute-solute enthalpic interaction parameters.
\[\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \nonumber \]
The corresponding excess isobaric heat capacity is defined by equation (b).
\[C_{p}^{E}\left(a q ; w_{1}=1 k g\right)=c_{p i j} \,\left(m_{j} / m^{0}\right)^{2} \nonumber \]
where
\[\mathrm{c}_{\mathrm{pij}}=\left(\frac{\partial \mathrm{h}_{\mathrm{ij}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
Here \(\mathrm{c}_{\mathrm{pjj}}\) is a pairwise solute-solute interaction isobaric heat capacity [1]. From
\[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\ell)-\mathrm{M}_{1} \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \nonumber \]
then,
\[\mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})=\mathrm{C}_{\mathrm{pl}^{2}}^{*}(\ell)-\mathrm{M}_{\mathrm{l}} \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \nonumber \]
From
\[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}^{\prime}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
then,
\[\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+2 \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Footnote
[1] For a solution prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of solute (at fixed \(\mathrm{T}\) and \(\mathrm{p}\))
\[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}(\mathrm{aq}) \nonumber \]
Hence
\[\begin{gathered}
\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{C}_{\mathrm{pl}}^{*}(\ell)-\mathrm{M}_{1} \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\right] \\
+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+2 \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\right]
\end{gathered} \nonumber \]
Then,
\[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\mathrm{c}_{\mathrm{pji}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \nonumber \]
Since,
\[\begin{gathered}
\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl} 1}^{*}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq}) \\
\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}
\end{gathered} \nonumber \]